Colour palette for PowerPoint presentations Primary colour ......Colour palette for PowerPoint presentations Actuarial Bright Green R148 G166 B31 Actuarial Slate R32 G44 B52 * Secondary

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Actuarial Bright Green R148 G166 B31

Actuarial Slate R32 G44 B52 *

Secondary Olive Green R120 G162 B47

Secondary colour palette

Primary colour palette

Secondary Bottle Green R0 G147 B127

Secondary Turquoise R0 G138 B176

Secondary Aqua Blue R26 G160 B170

Secondary Pastel Green R126 G205 B195

Secondary Light Purple R123 G149 B174*

Secondary Purple R97 G107 B156

Secondary Ecru R186 G163 B171

Secondary Yellow R215 G176 B18

Secondary Orange R213 G135 B43

Secondary Red R238 G52 B36

Secondary Rubine Red R226 G1 B119

*This colour reference is for screen presentations only

A better Bootstrap, Mack, and the ELRF and PTF modelling Frameworks •  Bootstrap technique- a powerful diagnostic tool for testing a

model; •  The Bootstrap is a technique not a model; •  When is the Bootstrap technique needed or necessary? •  Bootstrap samples (are supposed to) replicate the statistical

features of the real loss development (array); •  Two Families of models:

o Extended Link Ratio Family (ELRF) that includes Mack, Murphy and extensions/derivatives thereof;

o Probabilistic Trend Family (PTF) that fit a distribution to every cell, equivalently fit the trends in the three directions and the quality of the volatility about the trend structure

1



















Summary- Link Ratio Methods including Mack and relatives thereof

•  Link ratio methods - Mack & Murphy & quasi-Poisson GLM are structure-less, information free, no descriptors of the features in the data. Give incorrect calendar period liability stream;

•  On updating, estimates of mean ultimates may be grossly inconsistent;

•  Bootstrap samples generated from Mack method are easily distinguishable from the real data;

•  Mack, equivalently, volume weighted average (CL) link ratios do not distinguish between development and accident periods! It’s the same arithmetic irrespective of the statistical features in the data;

2



















Summary

•  PTF (and MPTF) modeling framework for building single-/multi-triangle models that can capture trend structure and volatility in real data- the latter also the three types of correlations

•  Identified model in PTF framework describes the trend structure and volatility succinctly (four pictures). All assumptions tested and validated.

•  Model satisfies axiomatic trend properties of every real datset •  Real loss triangle can be regarded as sample path from fitted

probabilistic model. Can’t tell the difference between real and simulated triangles. Also Bootstrap samples are indistinguishable from the real data

3



















Summary

•  Two LOBs written by the same company rarely have the same trend structure (including in the calendar year direction) and often process (volatility) correlation is either zero or very low. Reserve distribution correlation is often zero and if significant quite low.

•  No two companies are the same in respect of trend structure, and process (volatility) correlation is often zero (for the ‘same’ LOB).

•  No company is the same as the industry, unless it is a very large proportion of the industry.

•  All the above are demonstrated with real life data.

4



















Summary- Advantages of the PTF and MPTF modelling frameworks

•  Readily obtain percentiles , V@R and T-V@R tables for total reserve and aggregates, by calendar year and accident year for the aggregate of multiple LOBs and each LOB, conditional on explicit auditable assumptions

•  Measurement of the three types of correlations (relationships) between LOBs •  Obtain consistent estimates of prior year ultimates, and SII and IFRS 4

metrics on updating •  Calendar year liability stream distributions (and their correlations) are critical

for risk capital allocation and cost of capital calculations; and SII and IFRS 4 metrics (What do they depend on?)

•  Pricing future underwriting years •  No two companies are the same in respect of volatility and correlations

5



















Variability and Uncertainty

•  different concepts; not interchangeable

“Variability is a phenomenon in the physical world to be measured, analyzed and where appropriate explained. By contrast uncertainty is an aspect of knowledge.” –  Sir David Cox

6



















Example: Coin vs Roulette Wheel

Where do you need more risk capital? Introduce uncertainty into our knowledge - if coin or roulette wheel are mutilated then conclusions could be made only on the basis of observed data

7

Coin 100 tosses fair coin (#H?)

Mean = 50

Std Dev = 5

CI [50,50]

"Roulette Wheel" No. 0,1, …, 100 Mean = 50 Std Dev = 29 CI [50,50]

In 95% of experiments with the coin the number of heads will be in interval [40,60].

In 95% of experiments with the wheel, observed number will be in interval [2, 97].

… 1 0 1 0 0



















ELRF (Extended Link Ratio Family) Modelling Framework- Regression formulation of link ratios and extensions. Includes Mack, Murphy.

8

•

•• •

••

••

••

•

•

••

• •

}y{xX = Cum. @ j-1 Y = Cum. @ j

§  Link Ratios are a comparison of columns

j-1 j

y

x

§  We can graph the ratios of Y:X - line through O?

y/x y

x

y

x

y/x

Using ratios => E(Y|x) = βx

x is cumulative at dev. j-1 and y is cumulative at dev. j



















Mack (1993) is a regression formulation of volume weighted average link ratios

Chain Ladder Ratio (Volume Weighted Average)

9

( ) δσεε xVbxy 2 : =+=

∑∑=

∑

∑==

xy

xxyx

b̂ ,1 .1 δ

∑==xy

nb, . δ 1ˆ22

( )δx

wbxyw

1 where

2 Minimize=

−Σ

Arithmetic Average

1=δ



















IL(C) Data

Mack (=volume weighted average) weighted standardized residuals •  Note trend in residuals versus fitted values (bottom right)

10

Wtd Std Res vs Dev. Yr

0 1 2 3 4 5 6 7 8 9

-1.5-1

-0.5

0

0.5

1

1.52

Wtd Std Res vs Acc. Yr

81 82 83 84 85 86 87 88 89 90

-1.5-1

-0.5

0

0.5

1

1.52

Wtd Std Res vs Cal. Yr

81 82 83 84 85 86 87 88 89 90

-1.5-1

-0.5

0

0.5

1

1.52

Wtd Std Res vs Fitted

5,000 10,000 15,000 20,000 25,000

-1.5

-1

-0.5

0

0.5

1

1.5

2



















IL(C) Data Need intercepts- best link ratios are not through origin- hence method over fits big values and under fits small values

Cum.(1) vs Cum.(0)

0 1,000 2,000 3,000 4,000 5,000

01,0002,0003,0004,0005,0006,0007,0008,0009,00010,00011,00012,000

11

Cum.(2) vs Cum.(1)

0 2,000 4,000 6,000 8,000 10,000

0

2,000

4,000

6,000

8,000

10,000

12,000

14,000

16,000

18,000



















Intercept (Murphy (1994))

Since y already includes x: y = x + p, ie p = y - x

Incremental Cumulative at j at j -1 Is b -1 significant ? Venter (1996)

12

( )y a bx V x= + + =ε ε σ δ : 2

↑↑

( ) ( ) δσεε xVxbap 2 : 1 =+−+=



















Incr.(2) vs Cum.(1)

Corr. = 0.065, P-value = 0.8786,000 8,000 10,000

1,5002,0002,5003,0003,5004,0004,5005,0005,5006,0006,500

Incr.(1) vs Cum.(0)

Corr. = -0.117, P-value = 0.7641,000 2,000 3,000 4,000 5,000

2,5003,0003,5004,0004,5005,0005,5006,0006,5007,0007,5008,000

IL(C) data Link Ratios=1 in presence of an intercept. Zilch Predictive power

13

Incremental incurred not correlated to previous period cumulatives!



















Abandon Link Ratios - No predictive power

14

Case (ii) b a= ≠1 0

( ) ( ) δσεε xVxbap 2 : 1 =+−+=Case (i) b a> =1 0

( )a = Ave Incrementals

}y{x j-1 j j-1 j

} p

x x x x

x x

x

p Cumulative Incremental

Link ratio b has no predictive power





































Is assumption E(p | x ) = a + (b-1) x tenable?

•  Note: If corr(x, p) = 0, then corr((b-1)x, p) = 0 •  If x, p uncorrelated, no ratio has predictive power •  Ratio selection by actuarial judgment can’t overcome zero

correlation.

•  Corr. often close to 0 •  Sometimes not

–  Does this imply ratios are a good model?

–  Ranges?

15

p

x





































x

x x

x

x

x

x x

16

j-1 j

} p }y{x

j-1 j

w

90 91 92

Cumulative Incremental

Condition 1: p

w

Condition 2:

( ) ( ) δσεε xVxbap 2 : 1 =+−+=

Extended Link Ratio Family (ELRF) Modelling Framework



















Now Introduce Trend Parameter For Incrementals

17

12

n

w

{x }y

( ) ε+−++= xbwaap 110

a0 = Intercepta1 = Trendb = Ratio

p

p vs acci. yr, and previous cumulative



















The Probabilistic Trend Family (PTF) Modelling Framework Study in later slides

18

Condition 3: Incremental

Review 3 conditions:

Condition 1: Zero trend

Condition 2: Constant trend, positive or negative

Condition 3: Non-constant trend



















Mack=Chain Ladder (volume weighted average) treats accident years like development years Can cumulate across or down. Does not matter!

19

Dev per

Acci per

ratios across

Dev per

Acci per

cumulate across

cumulate down

ratios down project down

incremental array

2:

1:

project across



















Mack does not distinguish between accident years and development years

20

αγβ

αβα

γ =⎟⎠

⎞⎜⎝

⎛ −+

= 1p

αγβ

αγα

β =⎟⎠

⎞⎜⎝

⎛ −+

= 1p

The standard deviations are different because of different conditioning



















Dataset ABC- Worker’s Comp large company

Log-Normalised vs Dev. Year

0 1 2 3 4 5 6 7 8 9 10

9

9.5

10

10.5

11

11.5

12

21

Data versus development year Log-Normalised vs Acc. Year

77 78 79 80 81 82 83 84 85 86 87

9

9.5

10

10.5

11

11.5

12

Data versus accident year

Very different structure. So CL (Mack) ignores this information that sticks out!



















The Probabilistic Trend Family (PTF)Modelling Framework Here I will use the highlighter to illustrate rudimentary concepts

22

No Need for BF



















The PTF Modelling Framework Trend axioms satisfied by every real incremental triangle

•  Trends occur in three directions:

23

1 0 d

t = w+d

Development year

Accident year w

1986

1987

1998



















M3IR5 Data- Deterministic data with a single development period trend

100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 11080 9072 7427 100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 11080 9072

100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 11080

100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534

100000 81873 67032 54881 44933 36788 30119 24660 20190 16530

100000 81873 67032 54881 44933 36788 30119 24660 20190 100000 81873 67032 54881 44933 36788 30119 24660

100000 81873 67032 54881 44933 36788 30119

100000 81873 67032 54881 44933 36788

100000 81873 67032 54881 44933

100000 81873 67032 54881 100000 81873 67032

100000 81873

100000

24

0 1 2 3 4 5 6 7 8 9 10 11 12 13

α - 0.2d

d

PAID LOSS = EXP(alpha - 0.2d)

-0.2 alpha = 11.513



















Probabilistic Modelling We introduce three calendar year trends Axiomatic Properties of Trends

25

0.1

0.3

0.15



















Resultant development year trends (and accident year trends)

26

hidden block



















Trends + randomness

27

hidden block



















MODEL DISPLAYS- four integral graphs Graph bottom right represents process variability

28

Dev. Yr Trends

0 1 2 3 4 5 6 7 8 9 10 11 12 13

-2.5

-2

-1.5

-1

-0.5

0

-‐ 0 .2 0 6 2+-‐ 0 .0 0 3 3

Acc. Yr Trends

78 79 80 81 82 83 84 85 86 87 88 89 90 91

10.5

11

11.5

12

12.5

Cal. Yr Trends

78 79 80 81 82 83 84 85 86 87 88 89 90 91

0

0.5

1

1.5

2

0 .0 8 7 3+-‐ 0 .0 2 0 9

0 .3 9 2 7+-‐ 0 .0 4 4 2

0 .14 4 6+-‐ 0 .0 0 4 6

MLE Standard Deviation vs Dev. Yr

0 1 2 3 4 5 6 7 8 9 10 11 12 130

0.010.020.030.040.050.060.070.080.090.1



















Normal distribution about trend structure - integral part of model

Wtd Res Normality Plot

N = 105, P-value = 0.3867, R^2 = 0.9878-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-0.25-0.2

-0.15-0.1

-0.050

0.050.1

0.150.2

29



















Validation analyses- removal of years

Forecast Means and Standard Deviations vs Last Calendar Period

1 Unit = $11987 1988 1989 1990 1991

22,500,00023,000,00023,500,00024,000,00024,500,00025,000,00025,500,00026,000,00026,500,00027,000,00027,500,00028,000,00028,500,000

25,894,886+- 2,868,945

26,296,338+- 1,997,088

24,850,954+- 1,526,245

25,333,510+- 1,191,129

23,426,534+- 927,810

30

At end of 1991 Reserve dsn mean=23.4, SD=0.928, and at end 1987 mean=25.9, SD=2.87



















Forecast lognormals for each cell

•  All assumptions are explicit •  Process variability and parameter uncertainty included

31



















Simulate from forecast correlated lognormals Percentiles (Quantiles) and V@R statistics •  All assumptions are explicit •  Process variability and parameter uncertainty included

32


















*This colour reference is for screen presentations only 33

PROBABILISTIC MODEL

Real Data S1 S2

Simulated triangles cannot be distinguished from real data – similar trends, trend changes in same periods, same amount of random variation about trends

S3

Models project past volatility into the future

Trends+ variationabout trends



















Dataset ABC: The PTF model

Dev. Yr Trends

0 1 2 3 4 5 6 7 8 9 10

-3

-2.5

-2

-1.5

-1

-0.5

00 .16 6 1

+-‐ 0 .0 13 4

-‐ 0 .3 9 9 4+-‐ 0 .0 13 1

-‐ 0 .4 6 9 2+-‐ 0 .0 0 5 4

-‐ 0 .3 9 4 4+-‐ 0 .0 0 9 0

-‐ 0 .3 3 6 2+-‐ 0 .0 10 0

Acc. Yr Trends

77 78 79 80 81 82 83 84 85 86 87

9.5

10

10.5

11

11.5

12

12.5

13

0 .16 10+-‐ 0 .0 13 1

0 .0 4 7 3+-‐ 0 .0 10 5

0 .0 6 9 1+-‐ 0 .0 14 9

-‐ 0 .0 6 9 1+-‐ 0 .0 14 9

-‐ 0 .0 4 7 3+-‐ 0 .0 10 5

Cal. Yr Trends

77 78 79 80 81 82 83 84 85 86 87

-1

-0.5

0

0.5

1

1.5

2

0 .0 6 5 2+-‐ 0 .0 0 3 5

0 .10 8 3+-‐ 0 .0 12 3

0 .16 9 1+-‐ 0 .0 0 7 4

MLE Variance vs Dev. Yr

0 1 2 3 4 5 6 7 8 9 100

1e-4

2e-4

3e-4

4e-4

5e-4

6e-4

7e-4

8e-4

34 Note major calendar year trend shift

The optimal PTF identified model. Note the model fits a normal distribution to each cell. The means are related via the trend structure.



















Dataset ABC

•  As you move down the accident years the “kick-up” is one development period earlier

•  Real data satisfies axiomatic trend properties.

35

Log-Normalised vs Dev. Year

0 1 2 3 4 5 6 7 8 9 10

9

9.5

10

10.5

11

11.5

12



















Dataet ABC PTF-Calendar Year Trends Have control on future assumptions

36

Cal. Yr Trends

77 78 79 80 81 82 83 84 85 86 87

-1

-0.5

0

0.5

1

1.5

2

0 .0 6 5 2+-‐0 .0 0 3 5

0 .10 8 3+-‐0 .0 12 3

0 .16 9 1+-‐0 .0 0 7 4



















Dataset ABC: Three simulated triangles from the fitted model, and the real data triangle? Which is real data?

Dev. Yr Trends

0 1 2 3 4 5 6 7 8 9 10

-3

-2.5

-2

-1.5

-1

-0.5

0

0 .16 6 1+-‐ 0 .0 13 4

-‐ 0 .3 9 9 4+-‐ 0 .0 13 1

-‐ 0 .3 3 6 2+-‐ 0 .0 10 0

Acc. Yr Trends

77 78 79 80 81 82 83 84 85 86 879.5

10

10.5

11

11.5

12

12.5

13

0 .16 10+-‐ 0 .0 13 1

0 .0 4 7 3+-‐ 0 .0 10 5

-‐ 0 .0 6 9 1+-‐ 0 .0 14 9

Cal. Yr Trends

77 78 79 80 81 82 83 84 85 86 87

-1

-0.5

0

0.5

1

1.5

2

0 .0 6 5 2+-‐ 0 .0 0 3 5

0 .10 8 3+-‐ 0 .0 12 3

0 .16 9 1+-‐ 0 .0 0 7 4


0 1 2 3 4 5 6 7 8 9 100

1e-42e-43e-44e-45e-46e-47e-48e-4

37

Dev. Yr Trends

0 1 2 3 4 5 6 7 8 9 10

-3

-2.5

-2

-1.5

-1

-0.5

0

0 .16 9 6+-‐ 0 .0 12 2

-‐ 0 .3 9 7 6+-‐ 0 .0 119

-‐ 0 .3 3 0 3+-‐ 0 .0 0 9 0

Acc. Yr Trends

77 78 79 80 81 82 83 84 85 86 879.5

10

10.5

11

11.5

12

12.5

13

0 .16 0 5+-‐ 0 .0 119

0 .0 4 3 9+-‐ 0 .0 0 9 5

-‐ 0 .0 7 5 9+-‐ 0 .0 13 5

Cal. Yr Trends

77 78 79 80 81 82 83 84 85 86 87

-1

-0.5

0

0.5

1

1.5

2

0 .0 6 8 4+-‐ 0 .0 0 3 1

0 .0 9 8 9+-‐ 0 .0 111

0 .16 7 9+-‐ 0 .0 0 6 8


0 1 2 3 4 5 6 7 8 9 100

1e-4

2e-4

3e-4

4e-4

5e-4

6e-4

Dev. Yr Trends

0 1 2 3 4 5 6 7 8 9 10

-3

-2.5

-2

-1.5

-1

-0.5

0

0 .16 9 9+-‐ 0 .0 14 5

-‐ 0 .4 6 7 6+-‐ 0 .0 0 5 8

-‐ 0 .3 2 0 9+-‐ 0 .0 10 8

Acc. Yr Trends

77 78 79 80 81 82 83 84 85 86 87

10

10.5

11

11.5

12

12.5

13

0 .18 16+-‐ 0 .0 14 2

0 .0 2 3 7+-‐ 0 .0 114

-‐ 0 .0 8 5 1+-‐ 0 .0 16 1

Cal. Yr Trends

77 78 79 80 81 82 83 84 85 86 87

-1

-0.5

0

0.5

1

1.5

2

0 .0 6 0 1+-‐ 0 .0 0 3 7

0 .13 19+-‐ 0 .0 13 3


0 1 2 3 4 5 6 7 8 9 100

1e-42e-43e-44e-45e-46e-47e-48e-49e-4

Dev. Yr Trends

0 1 2 3 4 5 6 7 8 9 10

-3

-2.5

-2

-1.5

-1

-0.5

0

0 .15 5 2+-‐ 0 .0 0 9 6

-‐ 0 .3 9 0 9+-‐ 0 .0 0 9 4

-‐ 0 .3 3 6 5+-‐ 0 .0 0 7 2

Acc. Yr Trends

77 78 79 80 81 82 83 84 85 86 879.5

10

10.5

11

11.5

12

12.5

13

0 .15 9 1+-‐ 0 .0 0 9 5

0 .0 4 0 9+-‐ 0 .0 0 7 6

Cal. Yr Trends

77 78 79 80 81 82 83 84 85 86 87

-1

-0.5

0

0.5

1

1.5

2

0 .0 6 4 5+-‐ 0 .0 0 2 5

0 .10 19+-‐ 0 .0 0 8 8

0 .18 19+-‐ 0 .0 0 5 4


0 1 2 3 4 5 6 7 8 9 100

5e-51e-4

1.5e-42e-4

2.5e-43e-4

3.5e-44e-4



















Dataset ABC: Three simulated, one real. Residuals of fitting only one parameter in each direction. Which is the

real data? Simulated triangles have the same statistical features as the real data! We will use Bootstrap

technique later to do same thing.


0 1 2 3 4 5 6 7 8 9 10

-2-1.5-1

-0.50

0.51


77 78 79 80 81 82 83 84 85 86 87

-2-1.5-1

-0.50

0.51


77 78 79 80 81 82 83 84 85 86 87

-2-1.5-1

-0.50

0.51


9 10 11 12

-2-1.5

-1-0.5

00.5

1

38


0 1 2 3 4 5 6 7 8 9 10

-2-1.5-1

-0.50

0.51

1.5


77 78 79 80 81 82 83 84 85 86 87

-2-1.5-1

-0.50

0.51

1.5


77 78 79 80 81 82 83 84 85 86 87

-2-1.5-1

-0.50

0.51

1.5


9 10 11 12

-2-1.5-1

-0.50

0.51

1.5


0 1 2 3 4 5 6 7 8 9 10

-2-1.5-1

-0.50

0.51


77 78 79 80 81 82 83 84 85 86 87

-2-1.5-1

-0.50

0.51


77 78 79 80 81 82 83 84 85 86 87

-2-1.5-1

-0.50

0.51


9 10 11 12

-2-1.5-1

-0.50

0.51


0 1 2 3 4 5 6 7 8 9 10

-2-1.5-1

-0.50

0.51

1.5


77 78 79 80 81 82 83 84 85 86 87

-2-1.5-1

-0.50

0.51

1.5


77 78 79 80 81 82 83 84 85 86 87

-2-1.5-1

-0.50

0.51

1.5


9 10 11 12

-2-1.5-1

-0.50

0.51

1.5



















Dataset ABC- Wtd Standardized Residuals of Mack method (CL link ratios)


0 1 2 3 4 5 6 7 8 9 10

-1

-0.5

0

0.5

1

1.5


77 78 79 80 81 82 83 84 85 86 87

-1

-0.5

0

0.5

1

1.5


77 78 79 80 81 82 83 84 85 86 87

-1

-0.5

0

0.5

1

1.5


200,000 300,000 400,000 500,000

-1

-0.5

0

0.5

1

1.5

39

It is impossible for any link ratio method including Mack (=CL ratios) to capture and describe trends in any direction, let alone the calendar years.



















Dataset ABC ELRF- Mack (volume weighted average link ratios) Residuals versus calendar year. Cannot capture calendar year trend structure. No control on assumptions going forward either, and averager calendar year trend captured cannot be discerned.

40


77 78 79 80 81 82 83 84 85 86 87

-1

-0.5

0

0.5

1

1.5

Mack Residuals Cal. Yr Trends

77 78 79 80 81 82 83 84 85 86 87

-1

-0.5

0

0.5

1

1.5

2

0 .0 6 5 2+-‐ 0 .0 0 3 5

0 .10 8 3+-‐ 0 .0 12 3

0 .16 9 1+-‐ 0 .0 0 7 4

Left) Residuals after applying Mack method to the loss array for Dataset ABC. Note the sharp trend after 1984. Mack under fits recent calendar years and overfits earlier years. (Right) Probability Trend Family model picks up the change in trend structure in this direction, the other two directions and the volatility.

Calendar Year trends in incrementals



















Dataset ABC- Removing the three calendar year trends. That is setting the trend to zero for all calendar years in the PTF modelling framework

Looks a bit like the Mack residuals (but on a log scale)

41


77 78 79 80 81 82 83 84 85 86 87

-1

-0.5

0

0.5

1

1.5

2

2.5



















Dataset Mack (CL ratios) reserve too high by a factor of 2!


74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91

-1.5-1-0.500.511.522.5

42

Reserve = 901,941T +- 108,577T


74 76 78 80 82 84 86 88 90

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Reserve = 489,017T +_40,316T

Data trend minus trend estimated by Mack is negative

An ELRF model that better captures calendar year trend in recent cys



















The power of the PTF modelling framework

Cal. Yr Trends

89 90 91 92 93 94 95 96 97 98 99 00 01 02 03

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 .0 0 0 0+-‐0 .0 0 0 0

-‐0 .2 8 4 6+-‐0 .0 2 2 2

0 .0 6 8 8+-‐0 .0 2 4 9

0 .2 3 9 4+-‐0 .0 2 6 4

0 .0 6 8 8+-‐0 .0 2 4 9

43

COMPANY XYZ: CREs versus Paids. When was the company sold?

Cal. Yr Trends

89 90 91 92 93 94 95 96 97 98 99 00 01 02 03

-2.5-2-1.5-1-0.500.511.522.533.544.55

0 .2 9 5 8+-‐ 0 .0 15 9

0 .0 9 6 4+-‐ 0 .0 0 5 7

0 .2 9 5 8+-‐ 0 .0 15 9

CREs Paids



















The Bootstrap Technique- it is not a model! The Bootstrap can be used as a powerful diagnostic tool According to François Morin: "Bootstrapping utilizes the sampling-with-replacement technique

on the residuals of the historical data",

and

"Each simulated sampling scenario produces a new realization of "triangular data" that has the same statistical characteristics as the actual data." (Emphasis added)

- François Morin , Integrating Reserve Risk Models into Economic Capital Models, CLRS Seminar, Washington D.C. 2008

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This is worth repeating

•  "Each simulated sampling scenario produces a new realization of "triangular data" that has the same statistical characteristics as the actual data." (Emphasis added)

•  This only true if the model has the same statistical features as the data!

•  Bootstrap samples are generated from a model

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Bootstrap Samples

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Model

Bootstrap samples generated from model Data

BS1

…BS2 BS3 Real Data



















Do you Bootstrap a triangle? The observations in a triangle are not iid

47





































Bootstrapping the data is like assuming each fitted value is zero. That is, a residual = observation


0 1 2 3 4 5 6 7 8 9 10

0.8

0.85

0.9

0.95

1

1.05

1.1


0 1 2 3 4 5 6 7 8 9 10

0.80.820.840.860.880.90.920.940.960.981

1.021.041.061.081.11.12

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A bootstrap sample Data

You can easily tell the difference between the BS sample and the real data. So we need a better model

Would anybody want to do that? Why not?



















The Residuals

•  These are the differences between the observed values and the fitted values:

•  The residuals represent the trends in the data minus the trends estimated by the model.

49



















Bootstrapped Dataset

•  Working backwards from the bootstrapped residuals we form a bootstrap dataset

50

Data = Fit + residual

Bootstrap sample = Fit + re-sample residual (scaled)



















Bootstrap sample for a loss development array

data = fit + residual

=

Bootstrap data = fit + resample

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y ŷ r resample whole array of “Std residuals”

y* ŷ r*

1 2

3

Usually, r’s scaled to constant variance at step (2) then rescaled at step (3)

r*



















Mack and the bootstrap (Dataset ABC) The bootstrap as a diagnostic tool

•  Mack fitted to the real data contains structure by calendar year

Bootstrap samples from the Mack method lose this structure as it has been randomized!

52

Data



















Log-Linear Poisson Residuals versus Mack Residuals- very different. It is not the same model!

53

The Log-Linear Poisson residuals for Dataset ABC also show obvious structure in the calendar direction.


77 78 79 80 81 82 83 84 85 86 87

-1

-0.5

0

0.5

1

1.5

Mack residuals



















Dataset ABC: The optimal identified PTF model

•  The optimal PTF model for ABC (again)

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Dev. Yr Trends

0 1 2 3 4 5 6 7 8 9 10

-3

-2.5

-2

-1.5

-1

-0.5

00 .16 6 1

+-‐ 0 .0 13 4

-‐ 0 .3 9 9 4+-‐ 0 .0 13 1

-‐ 0 .4 6 9 2+-‐ 0 .0 0 5 4

-‐ 0 .3 9 4 4+-‐ 0 .0 0 9 0

-‐ 0 .3 3 6 2+-‐ 0 .0 10 0

Acc. Yr Trends

77 78 79 80 81 82 83 84 85 86 87

9.5

10

10.5

11

11.5

12

12.5

13

0 .16 10+-‐ 0 .0 13 1

0 .0 4 7 3+-‐ 0 .0 10 5

0 .0 6 9 1+-‐ 0 .0 14 9

-‐ 0 .0 6 9 1+-‐ 0 .0 14 9

-‐ 0 .0 4 7 3+-‐ 0 .0 10 5

Cal. Yr Trends

77 78 79 80 81 82 83 84 85 86 87

-1

-0.5

0

0.5

1

1.5

2

0 .0 6 5 2+-‐ 0 .0 0 3 5

0 .10 8 3+-‐ 0 .0 12 3

0 .16 9 1+-‐ 0 .0 0 7 4


0 1 2 3 4 5 6 7 8 9 100

1e-4

2e-4

3e-4

4e-4

5e-4

6e-4

7e-4

8e-4



















Mack bootstrap sample versus bootstrap samples from the identified PTF model (ABC)- The bootstrap technique as a diagnostic tool

Statistical CL applied to four datasets: Real, a Mack bootstrap sample, and

two bootstrap samples from the identified PTF model? No prize for guessing the odd man out!

55



















Residuals of fitting the model with a single parameter in each direction for three datasets: real and two BSs from the identified optimal PTF model

•  Which display is the real data? Impossible to tell!

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